Algebraic Springer-Verlag Berlin Heidelberg GmbH H. Koch

Algebraic Number Theory

Springer Consulting Editors of the Series: A.A. Agrachev, A.A. Gonchar, E.E Mishchenko, N.M. Ostianu, V.P. Sakharova, A.B. Zhishchenko

Title of the Russian edition: Itogi nauki i tekhniki, Sovremennye problemy matematiki, FundamentaPnye napravleniya, Vol. 62, Teoriya chisel 2 Publisher VINITI, Moscow 1988

Second Printing 1997 of the First Edition 1992, which was originally published as Number Theory II, Volume 62 of the Encyclopaedia of Mathematical Sciences.

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Koch, H: / H. Koch. Ed.: A. N. Parshin; I. R. Shafarevich. - 1. ed., 2. printing. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hongkong; London; Mailand; Paris; Santa Clara; Singapur; Tokio: Springer, 1997 ISBN 978-3-540-63003-6 ISBN 978-3-642-58095-6 (eBook) DOI 10.1007/978-3-642-58095-6

Mathematics Subject Classification (1991): nRxx, nSxx

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is per• mitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Originally published by Springer-Verlag Berlin Heidelberg New York in 1997 Softcover reprint of the hardcover 1st edition 1997

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SPIN: 10889896 41/3111-5432 1 - Printed on acid-free paper. List of Editors, Authors and Translators

Editor-in-Chief

R.V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Institute for Scientific Information (VINITI), ul. Usievicha 20 a, 125219 Moscow, Russia; e-mail: [email protected]

Consulting Editors

A. N. Parshin, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia 1. R. Shafarevich, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia

Author

H. Koch, Max-Planck-Gesellschaft zur Forderung der Wissenschaften e.V., Arbeitsgruppe "Algebraische Geometrie und Zahlentheorie" an der Humboldt-UniversiHit zu Berlin, JagerstraBe 10-11,10117 Berlin, Germany Algebraic Number Fields H. Koch

Contents

Preface...... 7 Chapter 1. Basic Number Theory ...... 8 § 1. Orders in Algebraic Number Fields ...... 9 1.1. Modules and Orders ...... 10 1.2. Module Classes ...... 12 1.3. The Group of an Order ...... 15 1.4. The Unit Group of a Real-Quadratic Number ...... 17 1.5. Integral Representations of Rational Numbers by Complete Forms...... 18 1.6. Binary Quadratic Forms and Complete Modules in Quadratic Number Fields...... 19 1.7. Representatives for Module Classes in Quadratic Number Fields...... 21 § 2. Rings with Divisor Theory ...... 22 2.1. Unique Factorization in Prime Elements ...... 22 2.2. The Concept of a Domain with Divisor Theory ...... 23 2.3. Divisor Theory for the Maximal Order of an ...... 25 § 3. Dedekind Rings ...... 27 3.1. Definition of Dedekind Rings ...... 28 3.2. Congruences ...... 29 3.3. Semilocalization ...... 30 3.4. Extensions of Dedekind Rings ...... 30 3.5. Different and Discriminant ...... 33 3.6. Inessential Discriminant Divisors ...... 36 3.7. Normal Extensions ...... 36 3.8. Ideals in Algebraic Number Fields ...... 40 3.9. Cyclotomic Fields ...... 41 3.10. Application to Fermat's Last Theorem I ...... 43 2 Contents

§ 4. Valuations ...... 45 4.1. Definition and First Properties of Valuations ...... 45 4.2. Completion of a Field with Respect to a ...... 49 4.3. Complete Fields with Discrete Valuation ...... 49 4.4. The Multiplicative Structure of a p-adic Number Field ...... 51 4.5. Extension of Valuations ...... 53 4.6. Finite Extensions of p-adic Number Fields ...... 55 4.7. Kummer Extensions ...... 58 4.8. Analytic Functions in Complete Non-Archimedean Valued Fields ...... 59 4.9. The Elementary Functions in p-adic Analysis ...... 60 4.10. Lubin-Tate Extensions ...... 62 § 5. Harmonic Analysis on Local and Global Fields ...... 63 5.1. Harmonic Analysis on Local Fields, the Additive Group .... 64 5.2. Harmonic Analysis on Local Fields, the Multiplicative Group 65 5.3. Adeles ...... 66 5.4. Ideles ...... 68 5.5. Subgroups of J(K)/Kx of Finite Index and the Ray Class Groups...... 69 § 6. Heeke L-Series and the Distribution of Prime Ideals ...... 70 6.1. The Local Zeta Functions ...... 74 6.2. The Global Functional Equation ...... 76 6.3. Heeke Characters ...... 77 6.4. The Functional Equation for Heeke L-Series ...... 79 6.5. Gaussian Sums ...... 80 6.6. Asymptotical Distribution of Ideals and Prime Ideals ...... 82 6.7. Chebotarev's Density Theorem ...... 84 6.8. Kronecker Densities and the Theorem of Bauer ...... 85 6.9. The Prime Ideal Theorem with Remainder Term ...... 87 6.10. Explicit Formulas ...... 87 6.11. Discriminant Estimation ...... 88

Chapter 2. ...... 90 § 1. The Main Theorems of Class Field Theory ...... 92 1.1. Class Field Theory for Abelian Extensions of Q ...... 92 1.2. The Hilbert Class Field ...... 93 1.3. ...... 93 1.4. The Idele Class Group of a Normal Extension ...... 95 1.5. Global Class Field Theory ...... 96 1.6. The Functorial Behavior of the Norm Symbol ...... 97 1.7. Artin's General ...... 98 1.8. The Power Residue Symbol ...... 99 1.9. The Hilbert Norm Symbol ...... 101 1.10. The Reciprocity Law for the Power Residue Symbol ...... 102 Contents 3

1.11. The Principal Ideal Theorem ...... 103 1.12. Local-Global Relations ...... 104 1.13. The Zeta Function of an ...... 105 § 2. Complex Multiplication ...... 107 2.1. The Main Polynomial ...... 107 2.2. The First Main Theorem ...... 108 2.3. The Reciprocity Law ...... 109 2.4. The Construction of the Ray Class Field ...... 109 2.5. Algebraic Theory of Complex Multiplication ...... 111 2.6. Generalization...... 112 § 3. of Groups ...... 112 3.1. Definition of Cohomology Groups ...... 112 3.2. Functoriality and the Long Exact Sequence ...... 113 3.3. Dimension Shifting ...... 114 3.4. Shapiro's Lemma ...... 115 3.5. Corestriction ...... 115 3.6. The Transgression and the Hochschild-Serre-Sequence ...... 116 3.7. Cup Product ...... 117 3.8. Modified Cohomology for Finite Groups ...... 119 3.9. Cohomology for Cyclic Groups ...... 120 3.10. The Theorem of Tate ...... 121 § 4. Proof of the Main Theorems of Class Field Theory ...... 121 4.1. Application of the Theorem of Tate to Class Field Theory ... 121 4.2. Class Formations ...... 122 4.3. Cohomology of Local Fields ...... 124 4.4. Cohomology of Ideles and Idele Classes ...... 125 4.5. Analytical Proof of the Second Inequality ...... 129 4.6. The Canonical Class for Global Extensions ...... 130 § 5. Simple Algebras ...... 131 5.1. Simple Algebras over Arbitrary Fields ...... 131 5.2. The Reduced Trace and Norm ...... 132 5.3. Splitting Fields ...... 133 5.4. The ...... 133 5.5. Simple Algebras over Local Fields ...... 134 5.6. The Structure of the Brauer Group of an Algebraic Number Field ...... _...... 135 5.7. Simple Algebras over Algebraic Number Fields ...... 136 § 6. Explicit Reciprocity Laws and Symbols ...... 137 6.1. The Explicit Reciprocity Law ofShafarevich ...... 138 6.2. The Explicit Reciprocity Law of Briickner and Vostokov .... 139 6.3. Application to Fermat's Last Theorem II ...... 141 6.4. Symbols ...... 142 6.5. Symbols of p-adic Number Fields ...... 143 6.6. Tame and Wild Symbols ...... 144 6.7. Remarks about Milnor's K-Theory ...... 144 4 Contents

§ 7. Further Results of Class Field Theory ...... 145 7.1. The Theorem of Shafarevich-Weil ...... 145 7.2. Universal Norms ...... 145 7.3. On the Structure of the Ideal Class Group ...... 146 7.4. Leopoldt's Spiegelungssatz ...... 147 7.5. The Cohomology of the Multiplicative Group ...... 149

Chapter 3. Galois Groups ...... 150 § 1. Cohomology of Profinite Groups ...... 151 1.1. Inverse Limits of Groups and Rings ...... 151 1.2. Profinite Groups ...... 153 1.3. Supernatural Numbers ...... 154 1.4. Pro-p-Groups and p-Sylow Groups ...... 154 1.5. Free Profinite, Free Prosolvable, and Free Pro-p-Groups .... 154 1.6. Discrete Modules ...... 155 1.7. Inductive Limits in C ...... 156 1.8. Galois Theory ofInfinite Algebraic Extensions ...... 157 1.9. Cohomology of Profinite Groups ...... 159 1.1 O. Cohomological Dimension ...... 159 1.11. The Dualizing Module ...... 160 1.12. Cohomology of Pro-p-Groups ...... 161 1.13. Presentation of Pro-p-Groups by Means of Generators and Relations ...... 162 1.14. Poincare Groups ...... 164 1.15. The Structure of the Relations and the Cup Product ...... 165 1.16. Group Rings and the Theorem of Golod-Shafarevich ...... 166 § 2. of Local and Global Fields ...... 168 2.1. Examples of Galois Cohomology of Arbitrary Fields ...... 168 2.2. The of a ...... 169 2.3. The Maximal p-Extension of a Local Field ...... 171 2.4. The of a Local Field ...... 173 2.5. The Maximal with Given Ramification .. 175 2.6. The Maximal p-Extension with Given Ramification ...... 177 2.7. The Class Field Tower Problem ...... 180 2.8. Discriminant Estimation from Above...... 181 2.9. Characterization of an Algebraic Number Field by its Galois Group ...... 182 § 3. Extensions with Given Galois Groups ...... 182 3.1. Embedding Problems ...... 183 3.2. Embedding Problems for Local and Global Fields ...... 185 3.3. Extensions with Prescribed Galois Group of I-Power Order .. 186 3.4. Extensions with Prescribed Solvable Galois Group ...... 188 3.5. Extensions with Prescribed Local Behavior ...... 188 3.6. Realization of Extensions with Prescribed Galois Group by Means of Hilbert's Irreducibility Theorem ...... 190 Contents 5

Chapter 4. Abelian Fields 192 § 1. The Integers of an Abelian Field ...... 193 1.1. The Coordinates ...... 193 1.2. The Galois Module Structure of the of Integers of an Abelian Field ...... 194 §2. The Arithmetical Class Number Formula ...... 195 2.1. The Arithmetical Class Number Formula for Complex Abelian Fields ...... 195 2.2. The Arithmetical Class Number Formula for Real Quadratic Fields ...... 197 2.3. The Arithmetical Class Number Formula for Real Abelian Fields ...... 198 2.4. The Stickelberger Ideal ...... 200 2.5. On the p-Component of the Class Group of Q('pm) ...... 203 2.6. Application to Fermat's Last Theorem III ...... 205 § 3. Iwasawa's Theory of r-Extensions ...... 206 3.1. Class Field Theory of r-Extensions ...... 206 3.2. The Structure of A-Modules ...... 207 3.3. The p-Class Group of a r-Extension ...... 208 3.4. Iwasawa's Theorem ...... 209 §4. p-adic L-Functions ...... 211 4.1. The Hurwitz Zeta Function ...... 212 4.2. p-adic L-Functions ...... 213 4.3. Congruences for Bernoulli Numbers ...... 214 4.4. Generalization to Totally Fields ...... 215 4.5. The p-adic Class Number Formula ...... 215 4.6. Iwasawa's Construction of p-adic L-Functions ...... 216 4.7. The Main Conjecture ...... 218

Chapter 5. Artin L-Functions and Galois Module Structure...... 219 § 1. Artin L-Functions ...... 222 1.1. Representations of Finite Groups ...... 222 1.2. Artin L-Functions ...... 224 1.3. Cyclotomic Fields with Class Number 1 ...... 226 1.4. Imaginary-Quadratic Fields with Small Class Number ...... 227 1.5. The Artin Representation and the ...... 228 1.6. The Functional Equation for Artin L-Functions ...... 230 1.7. The Conjectures of Stark about Artin L-Functions at s = 0 .. 231 § 2. Galois Module Structure and Artin Root Numbers ...... 234 2.1. The Class Group of Z[G] ...... 235 2.2. The Galois Module Structure of Tame Extensions ...... 236 2.3. Further Results on Galois Module Structure ...... 236 6 Contents

Appendix 1. Fields, Domains, and Complexes 237 1.1. Extensions ...... 237 1.2. Galois Theory ...... 238 1.3. Domains ...... 238 1.4. Complexes ...... 239

Appendix 2. Quadratic Residues ...... 240

Appendix 3. Locally Compact Groups ...... 241 3.1. Locally Compact Abelian Groups ...... 241 3.2. Restricted Products ...... 243

Appendix 4. Bernoulli Numbers ...... 243

Tables ...... 245

References ...... 251

Author Index 263

Subject Index 266 Preface

The purpose of the present article is the description of the main structures and results of algebraic number theory. The main subjects of interest are the algebra• ic number fields. We included in this article mostly properties of general charac• ter of algebraic number fields and of structures related to them. Special results appear only as examples which illustrate general features of the theory. On the other hand important parts of algebraic number theory such as class field theory have their analogies for other types of fields like functions fields of one variable over finite fields. More general class field theory is part of a theory of fields of finite dimension, which is now in development. We mention these analogies and generalizations only in a few cases. From the application of algebraic number theory to problems about Diophantine equations we explain the application to the theory of complete forms, in particular binary quadratic forms (Chap. 1.1.5-7), and to Fermat's last theorem (Chap. 1.3.10, Chap. 2.6.4, Chap. 4.2.6). A part of algebraic number theory serves as basis science for other parts of mathematics such as arithmetic algebraic geometry and the theory of modular forms. This concerns our Chap. 1, Basic number theory, Chap. 2, Class field theory, and parts of Chap. 3, Galois cohomology. These domains are presented in more detail than other parts of the theory. In accordance with the principles of this encyclopedia we give in general no reference to the history of results and refer only to the final or most convenient source for the result in question. Many paragraphs and sections contain after the title a main reference which served as basis for the presentation of the material of the paragraph or section. Each chapter and some paragraphs contain a detailed introduction to their subject. Therefore we give here only some technical remarks: At the end of Theorems, Propositions, Lemmas, Proofs, or Exercises stands the sign 0 or lEI. The sign 0 means that the Theorem, Proposition, Lemma, or Exercise is easy to prove or that we gave a full sketch of the proof. The sign lEI means that one needs for the proof ideas which are not or only partially explained in this article. We use Bourbaki's standard notation such as 7L for the , Q for the field of rational numbers, IR for the field of real numbers, and C for the field of complex numbers. For~ny ring A with unit element, AX denotes the group of units, i.e. the group of invertible elements and A* denotes the set of elements distinct from O. Furthermore J.l.n denotes the group of roots of unity with order dividing n. For any set M we denote by IMI the cardinality of M. I am grateful to I.R. Shafarevich and A.N. Parshin who read a preliminary version of this book and proposed several important alterations and additions, to U. Bellack and B. Wiist for their help in the production of the final manuscript and to the staff of Springer-Verlag for their cooperation.

Helmut Koch